GUP Hawking fermions from MGD black holes

Roberto Casadio,1, 2, ∗ Piero Nicolini,3, † and Roldao da Rocha4, ‡

1Dipartimento di Fisica e Astronomia, Universita di Bologna, via Irnerio 46, 40126 Bologna, Italy

2INFN, Sezione di Bologna, viale B. Pichat 6, 40127 Bologna, Italy

3Frankfurt Institute of Advanced Studies (FIAS) and Institut fur Theoretische Physik,

Goethe Universitat, Frankfurt am Main, Germany

4Centro de Matematica, Computacao e Cognicao,

Universidade Federal do ABC, 09210-580, Santo Andre, Brazil.

We derive the Hawking spectrum of fermions emitted by a minimally geometric deformed

(MGD) black hole. The MGD naturally describes quantum effects on the geometry in the

form of a length scale related, for instance, to the existence of extra dimensions. The

dynamics of the emitted fermions is described in the context of the generalised uncertainty

principle (GUP) and likewise contains a length scale associated with the quantum nature

of space-time. We then show that the emission is practically indistinguishable from the

Hawking thermal spectrum for large black hole masses, but the total flux can vanish for

small and finite black hole mass. This suggests the possible existence of black hole remnants

with a mass determined by the two length scales.

∗ casadio@bo.infn.it† nicolini@fias.uni-frankfurt.de‡ roldao.rocha@ufabc.edu.br

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I. INTRODUCTION

The minimal geometric deformation (MGD) was originally proposed [1, 2] as a systematic

method to determine high-energy corrections to general relativistic (GR) spherically symmetric so-

lutions in the brane-world [2–5]. It was also used to study bulk effects on realistic stellar interiors [6]

and the hydrodynamics of black strings [7]. Recently, the MGD corrections to the gravitational

lensing was estimated in Ref. [8], and it was shown that the merging of MGD stars could be detected

by the eLISA/LIGO experiments more easily than their Schwarzschild counterparts [9]. Finally, it

was proposed that the MGD can be realised in analogue gravitational systems which can be studied

in laboratories [10]. In particular, in brane-world models [11], our Universe is a (codimension-1)

brane with tension σ and the MGD leads to a deformation of the Schwarzschild metric proportional

to a positive length scale ` ∼ σ−1. Quite interestingly, the MGD was also shown to apply to more

general departures from GR than those predicted within the extra-dimensional scenario [12], being

stable under small linear perturbations [13]. In the following, we shall therefore consider the scale

` ∼ `p as related to a generic departure from GR induced by quantum physics.

The most renown quantum effect that should occur around black holes is the Hawking evap-

oration [14]. There are many derivations of this effect, most of which just assume a classical

background geometry. One of such approaches is the tunnelling method [15–18], which has been

considered both for bosons and fermions [19, 20] in various types of black hole backgrounds. The

(closely related) WKB approximations have then been employed in order to calculate quantum cor-

rections to the Bekenstein-Hawking entropy, e.g. for the Schwarzschild black hole. The tunnelling

method was recently employed in order to compute the Hawking radiation spectrum due to dark

spinors [21, 22]. Concerning in particular the emission of spin-1/2 fermions, the Hawking radia-

tion was analysed as the tunnelling of Dirac particles through an event horizon, where quantum

corrections in the single particle action are proportional to the usual semiclassical contribution.

The effects of the spin of each type of spin-1/2 fermions were then shown to cancel out, due to the

isotropy of the emission.

We shall here study the Hawking radiation of fermions from MGD black holes, including the

quantum effects on the fermion dynamics predicted by the Generalised Uncertainty Principle

(GUP), which has been comprehensively explored in Refs. [23–36], being compatible with a unitary

description. Such effects are also characterised by a (minimum) length scale β ∼ `2p, and can lead

either to the formation of hot [35] or cold [37] black hole remnants, or sub-Planckian black holes

[38]. It is therefore interesting to compare the changes to the quantum dynamics encoded by β

3

with the changes related to the (quantum) MGD represented by `. Although Refs. [39–41] already

analysed the GUP effects, no associated MGD effects have been studied in this context. Hence, we

shall here present a more general picture that can cover each of the above cases, by simply tuning

the respectively relevant parameters. In particular, we shall compute the corrected Hawking flux

by means of the tunnelling method, and show that the quantum modifications essentially depend

on `, with β generating only sub-leading deviations (at least according to this approximation). The

choice of spinors as emitted particles is based on the fact that the Hawking radiation favors lower

spin, lighter particles and it is expected to be dominated by fermions [42]. As a result we can rely

on the formalism developed in [21, 22].

The paper is organised as follows: in the next Section, we will briefly review the MGD black

hole metric for which the tunnelling rate for fermions will be computed in Section III; conclusions

and comments are then summarised in Section IV.

II. THE MGD BLACK HOLE

We recall that, in the brane-world scenario, the Gauss-Codazzi projection of the five-dimensional

Einstein equations yields the effective four-dimensional Einstein equations [11],

Rµν −1

2Rgµν = 8πGN T

effµν − Λ gµν , (1)

where GN = `p/mp, with mp and `p the four-dimensional Planck mass and scale, respectively;

Rµν and R are the Ricci tensor and scalar of the four-dimensional metric; Λ is the cosmological

constant (which we shall neglect hereafter). The effective stress tensor in Eq. (1) contains the

matter energy-momentum tensor on the brane, the electric component of the Weyl tensor and the

projection of the bulk energy-momentum tensor onto the brane [11]. For static and spherically

symmetric metrics,

ds2 = −A(r) dt2 +dr2

B(r)+ r2

(dϑ2 + sin2 ϑ dϕ2

), (2)

the MGD provides a solution to Eqs. (1) by deforming the radial metric component of the corre-

sponding GR solution [3, 5]. For the GR Schwarzschild metric, and dismissing terms of order σ−2

or higher, one obtains [3]

A(r) = 1− 2GNM

r, (3a)

B(r) = A(r)

[1 +

2 `

2 r − 3GNM

], (3b)

4

where ` ∼ σ−1 is the length scale previously discussed in the Introduction and M the ADM mass.

There are two solutions for the equation B(r) = 0, namely

r+ = 2GNM , (4a)

r− =3GNM

2− ` =

3

4r+ − ` , (4b)

so that r+ > r− for any ` > 0. For studying the Hawking radiation, we are interested in the region

outside r+, that effectively acts as the event horizon, and just note that r− is not a (Cauchy)

horizon [3].

We just mention in passing that an explicit expression for ` in terms of σ−1 can be obtained by

first considering a compact source of finite size r0 and proper mass M0 [1, 3], and then letting the

radius r0 decrease below r+. However, for practical purposes, it is more convenient and general

to show the dependence on the length `, as we mentioned in the Introduction. For example,

observational data impose bounds on the length `, from which bounds on σ can be straightforwardly

inferred according to the underlying model [43, 44].

III. HAWKING FLUX FOR FERMIONS

Let us now consider a GUP in the form 1

∆x∆p &~2

[1 + β∆p2

], (5)

where β = β0/m2p, and β0 is a (dimensionless) parameter encoding quantum gravity effects on the

particle dynamics. The upper bound β0 < 1021 was recently obtained (see [45, 46] and references

therein). In this framework, the position and momentum operators are respectively given by

xi = Xi and pi = Pi (1 + β p2), with i = 1, 2, 3. The variables Xi and Pi then satisfy the canonical

commutation relations [Xj , Pk] = i ~ δij , yielding

p2 = −~2 gij ∂i ∂j

(1− 2β ~2 gkl ∂

k∂l). (6)

The generalised frequency is defined by ω = E (1 − β E2) for E = i ~ ∂t. On the mass shell, the

energy of a particle with mass m and electric charge e reads [41, 47]

E = E[1 + β

(p2 +m2

)]. (7)

1 In our units, ~ = `pmp.

5

The Dirac equation in an external electromagnetic field Aµ (with µ = 0, . . . , 3) is given by

i γµ [~ (∂µ + Ωµ) + i eAµ] +mψ = 0 , (8)

where Ωµ ≡ i2 ω

αβµ Σαβ, and ω αβµ = −eρβ∂µe

αρ + eαν e

ρβΓνµρ is the spin connection. Eqs. (6) and (7)

can be replaced into Eq. (8), yielding the equation [41]i ~ γ0∂0 + [m− e γµ Aµ + i ~ γµ (Ωµ + ~β ∂µ)]

(1− β m2 + β ~2 gjk ∂

j ∂k)

ψ = 0 . (9)

The Hawking radiation emitted by black holes can contain several kinds of particles. Hereon,

we shall analyse the tunnelling of regular fermions across the event horizon of the MGD black

hole (3a)-(3b). The regular spinor field describing the fermion is assumed to be [21, 22]

Ψ =(ψ, 0, ψ, 0

)ᵀexp

i

~I(t, r, θ, φ)

, (10)

for an action I and wave-functions ψ and ψ. The metric (2) yields the tetrads

eαµ = diag

[√A(r),

1√B(r)

, r, r sin θ

], (11)

and the γµ matrices read

γt =1√A(r)

(i 0 0 00 i 0 00 0 −i 00 0 0 −i

), γθ =

1

r

(0 0 0 10 0 1 00 1 0 01 0 0 0

),

γr =√B(r)

(0 0 1 00 0 0 −11 0 0 00 −1 0 0

), γφ =

1

r sin θ

(0 0 0 −i0 0 i 00 −i 0 0i 0 0 0

). (12)

Inserting Eqs. (10) and (12) into the Dirac equation (9), the WKB approximation to leading order

in ~ yields the equations of motion

ψ

i√A

[∂tI − eAt

(1− β m2 − β K

)]−m

(1− β m2 + β K

)= ψ

(1− β m2 + β K

)√B ∂rI (13)

ψ

i√A

[∂tI + eAt

(1− β m2 − β K

)]+m

(1− β m2 + β K

)= −ψ

(1− β m2 + β K

)√B ∂rI (14)(

1− β m2 − β K)(

∂θI + i∂φI

sin θ

)= 0 , (15)

with

K = B (∂rI)2 +(∂θI)2

r2+

(∂φI)2

r2 sin2 θ. (16)

Upon writing the action in the usual form

I = −ω t+W (r) + Θ(θ, φ) , (17)

6

where ω is the energy of the emitted fermions, the tunnelling probability can now be derived [15,

18, 40]. Inserting Eq. (17) into Eq. (15), one obtains(∂φΘ

sin θ− i ∂θΘ

)[βB (W ′)2 + β

(∂θΘ)2

r2+ β

(∂φΘ)2

r2 sin2 θ+ βm2 − 1

]= 0 , (18)

where W ′ = dW/dr. Since the expression inside the square brackets cannot vanish, one must have

∂θΘ + i∂φΘ

sin θ= 0 , (19)

and the solution for Θ will therefore give no contribution to the tunnelling rate. Next, on sub-

stituting Eq. (17) with Eq. (19) into Eqs. (13) and (14), and again factoring out ψ and ψ, we

obtain

ψ0 + ψ2

(W ′)2

+ ψ4

(W ′)4

+ ψ6

(W ′)6

= 0 , (20)

where

ψ0 = −[m2A+ (eAt)

2] (

1− β m2)2 − ω2 + 2ω eAt

(1− β m2

), (21a)

ψ2 = β B

2 eAt

[eAt

(1− β m2

)− ω

]+A

(1− β2m4

), (21b)

ψ4 = −β B2[β (eAt)

2 + β A(2− β m2

)], (21c)

ψ6 = β2B3A . (21d)

Solving Eq. (20) on the event horizon yields the imaginary part of the action,

ImW±(r) = ±π4

r2+ ω (1 + β Ξ)

r+ − a r−, (22)

where

a =4M

(16M3/m3

p +M/mp + `/`p)

mp (3M/mp − 2 `/`p) (M/mp + `/`p), (23a)

Ξ =3

2m2 +

em2 At

ω − eAt− 2 e2 At (7M/mp − 2 `/`p)

M/mp + 2 `/`p+

2M ω

(M/mp + 2 `/`p). (23b)

Thus, the tunnelling rate of fermions reads

Γ ' exp(−2 ImΘ − 2 ImW+)

exp(−2 ImΘ − 2 ImW−)' exp

− 8πM2 (1 + β Ξ)ω

m3p (M/mp + `/`p)

, (24)

in which we just kept the leading order for ` . `pM/mp in the coefficient (23a). From now on, we

shall only consider neutral fermions (e = 0), so that

β Ξ = β0

[3m2

2m2p

+2M ω

m2p (M/mp + 2 `/`p)

], (25)

7

and the corresponding rate Γ is plotted in Figs. 1-5.

The rate (24) can be written as the Boltzmann-like factor Γ = exp (−ω/T ), where

T =m2

p (M + `0mp)

8πM2 (1 + β Ξ), (26)

and `0 = `/`p. We then observe that, for M mp (and ω ∼ m2p/M M), the above expression

can be approximated as

T ' T0

(1− 2β0

mp

M

)(27)

where

T0 =~

4π

[√A′(r)B′(r)

]r=r+

=m2

p

8πM

√1 +

2 `0mp

M'

m2p

8πM

(1 + `0

mp

M

)(28)

is the Hawking temperature of the MGD black hole obtained with the tunnelling method [19]

(and reduces to the standard Hawking expression for `0 = 0). The tunnelling rate (24) therefore

reproduces the Hawking result for large size black holes with a mass sufficiently large such that

both the GUP correction (proportional to β0) in Eq. (27) and the MGD correction (proportional

to `0) in Eq. (28) remain negligible.

For black hole mass approaching the Planck scale, the dependence of Ξ on ω ∼ M ∼ mp

cannot be neglected, and one cannot just consider the emission at a fixed temperature for all

frequencies. We can still assume the fermion mass m ' 0, since we consider M at least of the order

of the Planck scale, which is about 19 orders of magnitude heavier than the heaviest fundamental

fermions ever observed. We then regard Hawking particles with given energy ω in the angular

mode l. According to Eq. (24), these particles will be emitted with a probability approximately

given by the rate Γ(ω) = exp [−ω/T (ω)] multiplied by the probability for the black hole to absorb

such particles. Since the average number of fermions per mode,

nl(ω) =1

eω/T + 1=

Γ(ω)

Γ(ω) + 1, (29)

is related to the average emission rate nl(ω) = dnl(ω)/dt by 2π ~ nl(ω) = nl(ω) dω [42], the total

luminosity of the black hole can be obtained by multiplying by ω and summing over the modes,

that is

L =∑l=0

(2 l + 1)

∫|Tl (ω)|2 nl(ω)

ω dω

2π ~, (30)

where Tl (ω) are the grey-body factors. In the geometric optics regime, ω m2p/M , the sum over

the discrete angular modes l can be approximated with an integral and the luminosity then reads

L =T 4M2

2πm5p `p

∫ ∞0

(ωT

)3d(ωT

)∫ γ

0n

[(ωT

)(1 +

l (l + 1)m6p

128π2 T ωM4

)]d

(l (l + 1)m4

p

M2 ω2

), (31)

8

where the upper integration limit γ arises from studying wave scattering by the black hole. In this

respect, a black hole can be viewed as a black sphere, with an upper bound on the angular modes

that can be absorbed given by l (l + 1)m4p . 27M2 ω2 [39]. Since modes with l exceeding this

bound, for a given ω, are not absorbed by the black hole, they cannot be emitted either.

The flux (31) can indeed be computed with the complete expression for T in Eq. (26), but turns

out to be very cumbersome. For M mp, the leading behaviour is given by

L 'πm3

p

8 `pM2, (32)

and the evaporation rate M ' −L reproduces the expected Hawking expression for large black

holes, in agreement with the above analysis of the temperature T .

On the other hand, for M ' mp, the flux is given by

L(M, `0, β0,m) =πm3

p

8 `pM2+ β0

mp

`p

(πm2

p

35M2− M − 2mp

8mp

)− `0

mp

`p

(M2

m2p

+ F

), (33)

where F contains higher powers of M and of the parameters β0 and `0, but is rather involved

and we will just display its expansion to order M5/m5p for completeness below. The evaporation

rate M ' −L with the flux (33) again reproduces the Hawking expression for β0 = `0 = 0, but

can vanish for a finite value of M = Mc otherwise. The explicit expression of Mc is again very

complicated and we shall just estimate it in special regimes of the parameters `0 and β0. For

`0 = 0 and β0 > 0 the second term in Eq. (33) can be negative and compensate for the Hawking

contribution. In particular, for `0 = 0 and 0 < β0 1, the flux vanishes for

Mc(β0) ∼ β−1/30 mp , (34)

which can be rather large. Of course, β0 1 means that the length√β `p, and it is therefore

more sensible to consider β0 ' 1 for which we obtain

Mc ∼ 2.6mp . (35)

The above result does not change significantly for `0 = 0 and β0 1 (we recall that current bounds

on β0 are rather large [45, 46]), since the critical mass in this limit is given by

Mc ∼ 2.2mp . (36)

Conversely, when β0 = 0 and 0 < `0 1 (so that F is negligible), the third term in Eq. (33) leads

to the flux vanishing at a critical mass given by

Mc(`0) ∼ `−1/40 mp , (37)

9

which could again be fairly large. As for β0, it is physically more sensible to consider `0 ∼ 1, for

which the critical mass can be estimated numerically and is given by

Mc(`0) ∼ 2.1mp . (38)

We mentioned above that higher order (in M/mp) corrections to the flux are contained in the

function

F (M, `0, β0,m) '315π β2

0 `0m2p

M2+M2 `0m2

p

[2π

5

(3β0

m2

m2p

)− `0

16

]+M3 `20160m3

p

[5− 32π `30

(1+3β0

m2

m2p

)]+M4 `0m4

p

[`2010

(1+3β0

m2

m2p

)−(

1+3β0m2

m2p

)2

− `60π

]

+M5 `0

13440m5p

[105− 672π `30

(1+3β0

m2

m2p

)+2240π `70

(1+3β0

m2

m2p

)2], (39)

which will affect the precise value of the critical mass Mc. Let us note once more that the cor-

rection (39) vanishes for `0 = β0 = 0, as expected. Moreover, typical fermions emitted by black

holes will satisfy the bound β0m2 . 10−16m2

p and, since all terms of order O(M6) are multiplied

by factors of β20 m

4/m4p, the expression given above for the function F should be quite accurate for

quantum black holes.

IV. CONCLUDING REMARKS

The tunnelling method was here employed as a dynamical model of Hawking radiation in order

to compare the effects stemming from the MGD of the background black hole metric with those

described by a GUP applied to the quantum dynamics of the emitted particles. It is important to

stress that both the MGD and GUP corrections are of (quantum) gravitational origin, and such a

comparison could be useful for probing sectors of the semiclassical approach of quantum gravity.

Our calculations were based on the Hamilton-Jacobi method, which employs the WKB approx-

imation in order to solve the generalised Dirac equation, on a MGD background geometry. In

particular, we found that the total flux of fermions emitted by the black hole can vanish for a

critical mass (34)-(37), depending on the relative strength of the MGD parameter ` and the GUP

parameter β. Hence, black hole remnants, with a mass determined by these two length scales could

exist, say in the range between the critical masses (36) and (34) or (37), if also boson emission is

suppressed in the same regime. These remnants are also stable under small linear perturbations.

10

FIG. 1. Spectrum of the Hawking radia-

tion, as a function of the frequency ω and

black hole mass M (in powers of mp), for

β0 = 107 and `0 = 10−10.

FIG. 2. Spectrum of the Hawking radia-

tion, as a function of the frequency ω and

black hole mass M (in powers of mp), for

β0 = 0 = `0.

FIG. 3. Boltzmann factor, as a function

of the frequency ω and fermion mass m,

for β0 = 107 and `0 = 10−10.

FIG. 4. Spectrum of the Hawking radia-

tion, as a function of the frequency ω and

fermion mass m, for β0 = 0 = `0.

FIG. 5. Spectrum of the Hawking radiation, as a function of the frequency ω and GUP parameter β0, for `0 = 10−10.

In summary, the Hawking temperature of the MGD black hole is shown to be corrected by GUP

11

effects and the black hole evaporation is in general reduced by quantum effects of gravity.

ACKNOWLEDGMENTS

R.C. is partially supported by the INFN grant FLAG. The work of P.N. has been supported

by the project “Evaporation of the microscopic black holes” of the German Research Foundation

(DFG) under the grant NI 1282/2-2. R.dR. is grateful to CNPq (Grant No. 303293/2015-2), and

to FAPESP (Grant No. 2017/18897-8) and to INFN, for partial financial support.

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